Báo cáo hóa học: " Research Article Boundedness of the Maximal, Potential and Singular Operators in the Generalized Morrey Spaces"

Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009, Article ID 503948, 20 pages doi:10.1155/2009/503948 Research Article Boundedness of the Maximal, Potential and Singular Operators in the Generalized Morrey Spaces Vagif S Guliyev1, 2 Department of Mathematics, Ahi Evran University, Kirsehir, Turkey Institute of Mathematics and Mechanics, Baku, Azerbaijan Correspondence should be addressed to Vagif S Guliyev, vagif@guliyev.com Received 12 July 2009; Accepted 22 October 2009 Recommended by Shusen Ding We consider generalized Morrey spaces Mp,ω Rn with a general function ωx, r defining the Morrey-type norm We find the conditions on the pair ω1 , ω2 which ensures the boundedness of the maximal operator and Calderon-Zygmund singular integral operators from one generalized ´ Morrey space Mp,ω1 Rn to another Mp,ω2 Rn , < p < ∞, and from the space M1,ω1 Rn to the weak space WM1,ω2 Rn We also prove a Sobolev-Adams type Mp,ω1 Rn → Mq,ω2 Rn -theorem for the potential operators Iα In all the cases the conditions for the boundedness are given it terms of Zygmund-type integral inequalities on ω1 , ω2 , which not assume any assumption on monotonicity of ω1 , ω2 in r As applications, we establish the boundedness of some Schrodinger ă type operators on generalized Morrey spaces related to certain nonnegative potentials belonging to the reverse Holder class As an another application, we prove the boundedness of various ¨ operators on generalized Morrey spaces which are estimated by Riesz potentials Copyright q 2009 Vagif S Guliyev This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited Introduction For x ∈ Rn and r > 0, let Bx, r denote the open ball centered at x of radius r and Bx, r denote its complement n Let f ∈ Lloc R The maximal operator M, fractional maximal operator Mα , and the Riesz potential Iα are defined by Mfx sup|Bx, t|−1 t>0 f y dy, Bx,t Mα fx sup|Bx, t|−1α/n t>0 f y dy, Bx,t ≤ α < n, Journal of Inequalities and Applications Iα fx f y dy n−α , Rn x − y < α < n, 1.1 where |Bx, t| is the Lebesgue measure of the ball Bx, t Let T be a singular integral Calderon-Zygmund operator, briefly a Calderon-Zygmund operator, that is, a linear operator bounded from L2 Rn in L2 Rn taking all infinitely n continuously differentiable functions f with compact support to the functions T f ∈ Lloc R represented by T fx Rn K x, y f y dy 1.2 a.e on supp f Here Kx, y is a continuous function away from the diagonal which satisfies the standard estimates; there exist c1 > and < ε ≤ such that K x, y ≤ c1 x − y−n 1.3 y, and for all x, y ∈ Rn , x / K x, y − K x , y K y, x − K y, x ≤ c1 |x − x | x − y ε x − y−n , 1.4 whenever 2|x − x | ≤ |x − y| Such operators were introduced in 1 The operators M ≡ M0 , Mα , Iα , and T play an important role in real and harmonic analysis and applications see, e.g., 2, 3 Generalized Morrey spaces of such a kind were studied in 4–20 In the present work, we study the boundedness of maximal operator M and Calderon-Zygmund singular integral ´ operators T from one generalized Morrey space Mp,ω1 to another Mp,ω2 , < p < ∞, and from the space M1,ω1 to the weak space WM1,ω2 Also we study the boundedness of fractional maximal operator Mα and Riesz potential operators Mα from Mp,ω1 to Mq,ω2 , < p < q < ∞, and from the space M1,ω1 to the weak space WM1,ω2 , < q < ∞ As applications, we establish the boundedness of some Schodinger type operators on ă generalized Morrey spaces related to certain nonnegative potentials belonging to the reverse Holder class As an another application, we prove the boundedness of various operators on ă generalized Morrey spaces which are estimated by Riesz potentials Morrey Spaces In the study of local properties of solutions to of partial differential equations, together with weighted Lebesgue spaces, Morrey spaces Mp,λ Rn play an important role; see 21, 22 Journal of Inequalities and Applications Introduced by Morrey 23 in 1938, they are defined by the norm f Mp,λ : sup r −λ/p f Lp Bx,r , x,r>0 2.1 where ≤ λ < n, ≤ p < ∞ n We also denote by WMp,λ the weak Morrey space of all functions f ∈ WLloc p R for which f ≡ f WMp,λ Rn WMp,λ sup r −λ/p f WLp Bx,r < ∞, x∈Rn ,r>0 2.2 where WLp denotes the weak Lp -space Chiarenza and Frasca 24 studied the boundedness of the maximal operator M in these spaces Their results can be summarized as follows Theorem 2.1 Let ≤ p < ∞ and ≤ λ < n Then for p > the operator M is bounded in Mp,λ and for p M is bounded from M1,λ to WM1,λ The classical result by Hardy-Littlewood-Sobolev states that if < p < q < ∞, then Iα is bounded from Lp Rn to Lq Rn if and only if α n1/p − 1/q and for p < q < ∞, Iα is bounded from L1 Rn to WLq Rn if and only if α n1 − 1/q S Spanne published by Peetre 25 and Adams 26 studied boundedness of the Riesz potential in Morrey spaces Their results can be summarized as follows Theorem 2.2 Spanne, but published by Peetre 25 Let < α < n, independent of f such Iα f Mq,μ ≤ Cf Mp,λ 2.3 for every f ∈ Mp,λ Theorem 2.3 Adams 26 Let < α < n, independent of f such Iα f ≤ Cf Mp,λ Mq,λ 2.4 Mα fx ≤ υnα/n−1 Iα f x, 2.5 for every f ∈ Mp,λ Recall that, for < α < n, hence Theorems 2.2 and 2.3 also imply boundedness of the fractional maximal operator Mα , where is the volume of the unit ball in Rn The classical result for Calderon-Zygmund operators states that if < p < ∞ then T is bounded from Lp Rn to Lp Rn , and if p then T is bounded from L1 Rn to WL1 Rn see, e.g., 2 4 Journal of Inequalities and Applications Fazio and Ragusa 27 studied the boundedness of the Calderon-Zygmund singular ´ integral operators in Morrey spaces, and their results imply the following statement for Calderon-Zygmund operators T ´ Theorem 2.4 Let ≤ p < ∞, < λ < n Then for < p < ∞ Calderón-Zygmund singular integral operator T is bounded in Mp,λ and for p T is bounded from M1,λ to WM1,λ Note that in the case of the classical Calderon-Zygmund singular integral operators ´ Theorem 2.4 was proved by Peetre 25 If λ 0, the statement of Theorem 2.4 reduces to the aforementioned result for Lp Rn Generalized Morrey Spaces Everywhere in the sequel the functions ωx, r, ω1 x, r and ω2 x, r, used in the body of the paper are nonnegative measurable function on Rn × 0, ∞ We find it convenient to define the generalized Morrey spaces in the form as follows Definition 3.1 Let ≤ p < ∞ The generalized Morrey space Mp,ω Rn is defined of all n functions f ∈ Lloc p R by the finite norm r −n/p f Lp Bx,r x∈Rn ,r>0 ωx, r f sup Mp,ω 3.1 According to this definition, we recover the space Mp,λ Rn under the choice ωx, r r λ−n/p : Mp,λ Rn Mp,ω Rn |ωx,rr λ−n/p 3.2 In 4, 5, 17, 18 there were obtained sufficient conditions on weights ω1 and ω2 for the boundedness of the singular operator T from Mp,ω1 Rn to Mp,ω2 Rn In 18 the following condition was imposed on wx, r: c−1 ωx, r ≤ ωx, t ≤ c ωx, r, 3.3 whenever r ≤ t ≤ 2r, where c≥ 1 does not depend on t, r and x ∈ Rn , jointly with the condition ∞ ωx, tp r dt ≤ C ωx, rp , t 3.4 for the maximal or singular operator and the condition ∞ r tαp ωx, tp dt ≤ C r αp ωx, rp t 3.5 Journal of Inequalities and Applications for potential and fractional maximal operators, where C> 0 does not depend on r and x ∈ Rn Note that integral conditions of type 3.4 after the paper 28 of 1956 are often referred to as Bary-Stechkin or Zygmund-Bary-Stechkin conditions; see also 29 The classes of almost monotonic functions satisfying such integral conditions were later studied in a number of papers, see 30–32 and references therein, where the characterization of integral inequalities of such a kind was given in terms of certain lower and upper indices known as MatuszewskaOrlicz indices Note that in the cited papers the integral inequalities were studied as r → Such inequalities are also of interest when they allow to impose different conditions as r → and r → ∞; such a case was dealt with in 33, 34 In 18 the following statements were proved Theorem 3.2 18 Let ≤ p < ∞ and ωx, r satisfy conditions 3.3-3.4 Then for p > the operators M and T are bounded in Mp,ω Rn and for p M and T are bounded from M1,ω Rn to WM1,ω Rn Theorem 3.3 18 Let ≤ p < ∞, < α < n/p, 1/q 1/p−α/n and ωx, t satisfy conditions 3.3 and 3.5 Then for p > the operators Mα and Iα are bounded from Mp,ω Rn to Mq,ω Rn and for p Mα and Iα are bounded from M1,ω Rn to WMq,ω Rn The Maximal Operator in the Spaces Mp,ω Rn n Theorem 4.1 Let ≤ p < ∞ and f ∈ Lloc p R Then for p > Mf ≤ Ctn/p Lp Bx,t ∞ t r −n/p−1 f Lp Bx,r dr, 4.1 and for p Mf ≤ Ctn WL1 Bx,t ∞ t r −n−1 f L1 Bx,r dr, 4.2 where C does not depend on f, x ∈ Rn and t > Proof Let 0, 4.3 and have Mf ≤ Mf1 Lp Bx,t Mf2 Lp Bx,t Lp Bx,t 4.4 By boundedness of the operator M in Lp Rn , < p < ∞ we obtain Mf1 Lp Bx,t ≤ Mf1 Lp Rn ≤ Cf1 Lp Rn Cf Lp Bx,2t , 4.5 Journal of Inequalities and Applications where C does not depend on f From 4.5 we have Mf1 ≤ Ctn/p Lp Bx,t ≤ Ct n/p ∞ 2t r −n/p−1 f Lp Bx,r dr ∞ r 4.6 f Lp Bx,r dr −n/p−1 t easily obtained from the fact that f Lp Bx,2t is nondecreasing in t, so that f Lp Bx,2t on the right-hand side of 4.5 is dominated by the right-hand side of 4.6 To estimate Mf2 , we first prove the following auxiliary inequality: Bx,t x − y−n f y dy ≤ C ∞ t s−n/p−1 f Lp Bx,s ds, < t < ∞ 4.7 To this end, we choose β > n/p and proceed as follows: Bx,t x − y−n f y dy ≤ β x − y−nβ f y dy Bx,t ∞ β s−β−1 ds t ≤C ∞ {y∈Rn :t≤|x−y |≤s} ∞ |x−y| s−β−1 ds x − y−nβ f y dy −nβ s−β−1 f Lp Bx,s x − y Lp Bx,s t 4.8 ds For z ∈ Bx, t we get Mf2 z sup|Bz, r|−1 r>0 r≥2t ≤ Csup r≥2t ≤C f2 y dy Bz,r ≤ Csup Bx,2t∩Bz,r Bx,2t∩Bz,r Bx,2t y − z−n f y dy x − y−n f y dy 4.9 x − y−n f y dy Then by 4.7 Mf2 z ≤ C ∞ 2t ≤C ∞ s−n/p−1 f Lp Bx,s ds s t f Lp Bx,s ds, −n/p−1 4.10 Journal of Inequalities and Applications where C does not depend on x, r Thus, the function Mf2 z, with fixed x and t, is dominated by the expression not depending on z Then Mf2 ≤C Lp Bx,t ∞ t s−n/p−1 f Lp Bx,s ds 1 Lp Bx,t 4.11 Since 1 Lp Bx,t Ctn/p , we then obtain 4.1 from 4.6 and 4.11 Let p It is obvious that for any ball B Bx, r Mf ≤ Mf1 WL1 Bx,t Mf2 WL1 Bx,t WL1 Bx,t 4.12 By boundedness of the operator M from L1 Rn to WL1 Rn we have Mf1 WL1 Bx,t ≤ Cf L1 Bx,2t , 4.13 where C does not depend on x, t Note that inequality 4.11 also true in the case p Then by 4.11, we get inequality 4.2 Theorem 4.2 Let ≤ p < ∞ and the function ω1 x, r and ω2 x, r satisfy the condition ∞ ω1 x, r t dr ≤ C ω2 x, t, r 4.14 where C does not depend on x and t Then for p > the maximal operator M is bounded from Mp,ω1 Rn to Mp,ω2 Rn and for p 1M is bounded from M1,ω1 Rn to WM1,ω2 Rn Proof Let 0 ≤ C sup ω2−1 x, t ∞ x∈Rn , t>0 t r −n/p−1 f Lp Bx,r dr 4.15 Hence Mf ≤ Cf Mp,ω Mp,ω ≤ Cf Mp,ω sup x∈Rn , t>0 ω2 x, t by 4.14, which completes the proof for 0 ≤ C sup ω2−1 x, t ∞ x∈Rn , t>0 t r −n−1 f L1 Bx,r dr 4.17 Hence Mf ≤ Cf M1,ω WM1,ω ≤ Cf M1,ω n R x∈Rn , t>0 ω2 x, t ∞ sup ω1 x, r t dr r 4.18 by 4.14, which completes the proof for p Remark 4.3 Note that Theorems 4.1 and 4.2 were proved in 4 see also 5 Theorem 4.2 not impose the pointwise doubling conditions 3.3 and 3.4 In the case ω1 x, r ω2 x, r ωx, r, Theorem 4.2 is containing the results of Theorem 3.2 Riesz Potential Operator in the SpacesMp,ω Rn 5.1 Spanne Type Result n Theorem 5.1 Let ≤ p < ∞, < α < n/p, 1/q 1/p − α/n, and f ∈ Lloc p R Then for p > Iα f ≤ Ctn/q Lq Bx,t ∞ t r −n/q−1 f Lp Bx,r dr, 5.1 and for p Iα f ≤ Ctn/q WLq Bx,t ∞ t r −n/q−1 f L1 Bx,r dr, 5.2 where C does not depend on f, x ∈ Rn and t > Proof As in the proof of Theorem 4.1, we represent function f in form 4.3 and have Iα fx Iα f1 x Iα f2 x 5.3 Let < p < ∞, < α < n/p, 1/q 1/p − α/n By boundedness of the operator Iα from Lp Rn to Lq Rn we obtain Iα f1 Lq Bx,t ≤ Iα f1 Lq Rn ≤ Cf1 Lp Rn C f Lp Bx,2t 5.4 Journal of Inequalities and Applications Then Iα f1 ≤ Cf Lp Bx,2t , Lq Bx,t 5.5 where the constant C is independent of f Taking into account that f ≤ Ctn/q Lp Bx,2t ∞ 2t r −n/q−1 f Lp Bx,r dr, 5.6 we get Iα f1 ≤ Ctn/q Lq Bx,t ∞ 2t r −n/q−1 f Lp Bx,r dr 5.7 When |x − z| ≤ t, |z − y| ≥ 2t, we have 1/2|z − y| ≤ |x − y| ≤ 3/2|z − y|, and therefore α−n Iα f2 z − y fydy ≤ Lq Bx,t Bx,2t Lq Bx,t ≤C B x,2t 5.8 x − yα−n f y dyχBx,t Lq Rn We choose β > n/q and obtain Bx,2t |x − y| f y dy β α−n β ∞ x − yα−nβ f y Bx,2t ∞ −β−1 s 2t ≤C ∞ 2t ≤C ∞ 2t {y∈Rn :2t≤|x−y|≤s} |x−y| s −β−1 ds dy x − yα−nβ f y dy ds s−β−1 f Lp Bx,s |x − y|α−nβ Lp Bx,s 5.9 ds sα−n/p−1 f Lp Bx,s ds Therefore Iα f2 ≤ Ctn/q Lq Bx,t ∞ 2t s−n/q−1 f Lp Bx,s ds, 5.10 which together with 5.7 yields 5.1 Let p It is obvious that for any ball B Bx, r Iα f ≤ Iα f1 WL1 Bx,t Iα f2 WL1 Bx,t WL1 Bx,t 5.11 10 Journal of Inequalities and Applications By boundedness of the operator Iα from L1 Rn to WLq Rn we have Iα f1 ≤ Cf Lq Bx,2t , WL1 Bx,t 5.12 where C does not depend on x, t Note that inequality 5.10 also true in the case p Then by 5.10, we get inequality 5.2 Theorem 5.2 Let ≤ p < ∞, < α < n/p, 1/q 1/p − α/n and the functions ω1 x, r and ω2 x, r fulfill the condition ∞ tα ω1 x, t r dt ≤ C ω2 x, r, t 5.13 where C does not depend on x and r Then for p > the operators Mα and Iα are bounded from Mp,ω1 Rn to Mq,ω2 Rn and for p Mα and Iα are bounded from M1,ω1 Rn to WMq,ω2 Rn Proof Let 0 ω2 x, t ≤ Cf Mp,ω ∞ t r −n/q−1 f Lp Bx,r dr sup ω t x, x∈Rn , t>0 ∞ t 5.14 dr r ω1 x, r r α by 5.13, which completes the proof for 0 ≤ C sup ω2−1 x, t x∈Rn , t>0 ∞ t r −n/q−1 f L1 Bx,r dr 5.15 Hence Iα f ≤ Cf M1,ω WMq,ω ≤ Cf M1,ω sup Rn ω t x, n x∈R , t>0 ∞ t r α ω1 x, r dr r 5.16 by 5.13, which completes the proof for p Remark 5.3 Note that Theorems 5.1 and 5.2 were proved in 4 see also 5 Theorem 5.2 not impose the pointwise doubling condition, 3.3 and 3.5 In the case ω1 x, r ω2 x, r ωx, r, Theorem 5.2 is containing the results of Theorem 3.3 Journal of Inequalities and Applications 11 5.2 Adams Type Result n Theorem 5.4 Let ≤ p < ∞, < α < n/p, and f ∈ Lloc p R Then Iα fx ≤ Ctα Mfx C ∞ t r α−n/p−1 f Lp Bx,r dr, 5.17 where C does not depend on f, x, and t Proof As in the proof of Theorem 4.1, we represent function f in form 4.3 and have Iα fx Iα f1 x Iα f2 x 5.18 For Iα f1 x, following Hedberg’s trick see for instance 2, page 354, we obtain |Iα f1 x| ≤ C1 tα Mfx For Iα f2 x we have Iα f2 x ≤ ≤C ≤C Bx,2t Bx,2t ∞ 2t ≤C x − yα−n f y dy ∞ t f y dy ∞ |x−y| r α−n−1 dr f y dy r α−n−1 dr 5.19 2t such that ωx, · : 0, ∞ −→ a, ∞ is surjective 5.21 Then for p > the operators Mα and Iα are bounded from Mp,ω Rn to Mq,ωp/q Rn and for p the operators Mα and Iα are bounded from M1,ω Rn to WMq,ω1/q Rn Proof In view of the well-known pointwise estimate Mα fx ≤ CIα |f|x, it suffices to treat only the case of the operator Iα 12 Journal of Inequalities and Applications Let ≤ p < ∞ and f ∈ Mp,ω Rn By Theorem 5.4 we get Iα fx ≤ Cr α Mfx Cf Mp,ω ∞ tα ωx, t r dt t 5.22 From 5.20 we have r α ωx, r ≤ Cωx, rp/q Making also use of condition 5.20, we obtain Iα fx ≤ Cωx, rp/q−1 Mfx Cωx, rp/q f Mp,ω 5.23 Since ωx, r is surjective, we can choose r > so that ωx, r Mfx f −1 Mp,ω Rn , assuming n that f is not identical Hence, for every x ∈ R , we have Iα fx ≤ C Mfx p/q f 1−p/q Mp,ω 5.24 Hence the statement of the theorem follows in view of the boundedness of the maximal operator M in Mp,ω Rn provided by Theorem 4.2 in virtue of condition 4.14 Iα f M q,ωp/q sup ωx, t−p/q t−n/q Iα f Lq Bx,t x∈Rn , t>0 1−p/q ≤ C f Mp,ω ≤ C f Mp,ω , p/q sup ωx, t−p/q t−n/q Mf Lp Bx,t 5.25 x∈Rn , t>0 if 0 1−1/q ≤ C f M1,ω ≤ C f M1,ω , 1/q sup ωx, t−1/q t−n/q Mf WL1 Bx,t 5.26 x∈Rn , t>0 if p < q < ∞ Singular Operators in the Spaces Mp,ω Rn n Theorem 6.1 Let ≤ p < ∞ and f ∈ Lloc p R Then for p > T f ≤ Ctn/p Lp Bx,t ∞ t r −n/p−1 f Lp Bx,r dr, 6.1 Journal of Inequalities and Applications 13 and for p T f WL1 Bx,t ≤ Ctn ∞ t r −n−1 f L1 Bx,r dr, 6.2 where C does not depend on f, x ∈ Rn and t > Proof Let < p < ∞ We represent function f as in 4.3 and have T f ≤ T f1 Lp Bx,t T f2 Lp Bx,t Lp Bx,t 6.3 By boundedness of the operator T in Lp Rn , < p < ∞ we obtain T f1 Lp Bx,t ≤ T f1 Lp Rn ≤ C f1 Lp Rn , so that T f1 ≤ Cf Lp Bx,2t Lp Bx,t 6.4 Taking into account the inequality f Lp Bx,t ≤ Ct n/p ∞ 2t r −n/p−1 f Lp Bx,r dr, 6.5 we get T f1 ≤ Ctn/p Lp Bx,t ∞ 2t r −n/p−1 f Lp Bx,r dr 6.6 To estimate T f2 Lp Bx,t , we observe that T f2 z ≤ C f y dy n , Bx,2t y − z 6.7 where z ∈ Bx, t and the inequalities |x − z| ≤ t, |z − y| ≥ 2t imply 1/2|z − y| ≤ |x − y| ≤ 3/2|z − y|, and therefore T f2 Lp Bx,t ≤C x − y−n f y dyχBx,t Lp Rn Bx,2t 6.8 Hence by inequality 4.7, we get T f2 ≤ Ctn/p Lp Bx,t From 6.6 and 6.9 we arrive at 6.1 ∞ 2t r −n/p−1 f Lp Bx,r dr 6.9 14 Journal of Inequalities and Applications Let p It is obvious that for any ball Bx, r T f ≤ T f1 WL1 Bx,t T f2 WL1 Bx,t WL1 Bx,t 6.10 By boundedness of the operator T from L1 Rn to WL1 Rn we have T f1 ≤ Cf L1 Bx,2t , WL1 Bx,t 6.11 where C does not depend on x, t Note that inequality 6.9 also true in the case p Then by 4.11, we get inequality 6.2 Theorem 6.2 Let ≤ p < ∞ and ω1 x, t and ω2 x, r fulfill condition 4.14 Then for p > the singular integral operator T is bounded from the space Mp,ω1 Rn to the space Mp,ω2 Rn and for p T is bounded from M1,ω1 Rn to WM1,ω2 Rn Proof Let 0 ∞ −1 C sup ω2 x, t r −n/p−1 f Lp Bx,r dr n t x∈R , t>0 6.12 Hence T f ≤ Cf Mp,ω Mp,ω ≤ C f 1 ω t x, n x∈R , t>0 ∞ sup ω1 x, r t dr r 6.13 Mp,ω1 by 4.14, which completes the proof for 0 ≤ C sup ω2−1 x, t x∈Rn , t>0 ∞ t r −n−1 f L1 Bx,r dr 6.14 Journal of Inequalities and Applications 15 Hence T f ≤ Cf M1,ω WM1,ω ≤ C f sup Rn ω t x, n x∈R , t>0 ∞ ω1 x, r t dr r 6.15 M1,ω1 by 4.14, which completes the proof for p Remark 6.3 Note that Theorems 6.1 and 6.2 were proved in 4 see also 5 Theorem 6.2 does not impose the pointwise doubling conditions 3.3 and 3.4 In the case ω1 x, r ω2 x, r ωx, r, Theorem 6.2 is containing the results of Theorem 3.2 The Generalized Morrey Estimates for the Operators V γ −Δ V −β and V γ ∇−Δ V −β In this section we consider the Schrodinger operator −Δ V on Rn , where the nonnegative ¨ potential V belongs to the reverse Holder class B∞ Rn for some q1 n The generalized ă n Morrey Mp,ω R estimates for the operators V γ −Δ V −β and V γ ∇−Δ V −β are obtained The investigation of Schrodinger operators on the Euclidean space Rn with nonnegaă tive potentials which belong to the reverse Holder class has attracted attention of a number ă of authors cf 3537 Shen 36 studied the Schrodinger operator V , assuming the ă nonnegative potential V belongs to the reverse Holder class Bq Rn for q n/2 and he ă proved the Lp boundedness of the operators −Δ V iγ , ∇2 −Δ V −1 , ∇−Δ V −1/2 , and ∇−Δ V −1 Kurata and Sugano generalized Shens results to uniformly elliptic operators in 38 Sugano 39 also extended some results of Shen to the operator V γ −Δ V −β , ≤ γ ≤ β ≤ 1, and V γ ∇−Δ V −β , ≤ γ ≤ 1/2 ≤ β ≤ and β − γ ≥ 1/2 Later, Lu 40 and Li 41 investigated the Schrodinger operators in a more general setting ă We investigate the generalized Morrey Mp,1 -Mq,2 boundedness of the operators T1 V γ −Δ V −β , T2 V γ ∇−Δ V −β , ≤ γ ≤ β ≤ 1, 0≤γ ≤ 1 ≤ β ≤ 1, β − γ ≥ 2 7.1 Note that the operators V −Δ V −1 and V 1/2 ∇−Δ V −1 in 41 are the special case of T1 and T2 , respectively It is worth pointing out that we need to establish pointwise estimates for T1 , T2 and their adjoint operators by using the estimates of fundamental solution for the Schrodinger ă operator on Rn in 41 And we prove the generalized Morrey estimates by using Mp,ω1 −Mq,ω2 boundedness of the fractional maximal operators 16 Journal of Inequalities and Applications Let V ≥ We say V ∈ B∞ , if there exists a constant C > such that V L∞ B ≤ C |B| V xdx 7.2 B holds for every ball B in Rn see 41 The following are two pointwise estimates for T1 and T2 which are proven in 37, Lemma 3.2 with the potential V ∈ B∞ Theorem B Suppose V ∈ B∞ and ≤ γ ≤ β ≤ Then there exists a constant C > such that T1 fx ≤ CMα fx, f ∈ C0∞ Rn , 7.3 where α 2β − γ Theorem C Suppose V ∈ B∞ , ≤ γ ≤ 1/2 ≤ β ≤ and β − γ ≥ 1/2 Then there exists a constant C > such that T2 fx ≤ CMα fx, f ∈ C0∞ Rn , 7.4 where α 2β − γ − The previous theorems will yield the generalized Morrey estimates for T1 and T2 Corollary 7.1 Assume that V ∈ B∞ , and ≤ γ ≤ β ≤ Let ≤ p ≤ q < ∞, 2β−γ n1/p−1/q, and condition 5.13 be satisfied for α 2β − γ Then for p > the operator T1 is bounded from Mp,ω1 Rn to Mq,ω2 Rn and for p T1 is bounded from M1,ω1 Rn to WMq,ω2 Rn Corollary 7.2 Assume that V ∈ B∞ , ≤ γ ≤ 1/2 ≤ β ≤ 1, and β − γ ≥ 1/2 Let ≤ p ≤ q < ∞, 2β − γ − n1/p − 1/q, and condition 5.13 be satisfied for α 2β − γ − Then for p > the operator T2 is bounded from Mp,ω1 Rn to Mq,ω2 Rn and for p T2 is bounded from M1,ω1 Rn to WMq,ω2 Rn Some Applications The theorems of Section can be applied to various operators which are estimated from above by Riesz potentials We give some examples Suppose that L is a linear operator on L2 which generates an analytic semigroup e−tL with the kernel pt x, y satisfying a Gaussian upper bound, that is, pt x, y ≤ c1 e−c2 |x−y| /t tn/2 for x, y ∈ Rn and all t > 0, where c1 , c2 > are independent of x, y, and t 8.1 Journal of Inequalities and Applications 17 For < α < n, the fractional powers L−α/2 of the operator L are defined by L −α/2 fx Γα/2 ∞ e−tL fx dt t−α/21 8.2 Note that if L −Δ is the Laplacian on Rn , then L−α/2 is the Riesz potential Iα See, for example, 2, Chapter 5 Theorem 8.1 Let < α < n, ≤ p < q < ∞, α/n 1/p − 1/q and conditions 5.13, 8.1 are satisfied Then for p > the operator L−α/2 is bounded from Mp,ω1 Rn to Mq,ω2 Rn and for p 1L−α/2 is bounded from M1,ω1 Rn to WMq,ω2 Rn Proof Since the semigroup e−tL has the kernel pt x, y which satisfies condition 8.1, it follows that −α/2 8.3 fx ≤ CIα f x L for all x ∈ Rn , where C > is independent of x see 42 Hence by Theorem 5.2 we have −α/2 L f Mq,ω ≤ CIα f Mq,ω ≤ Cf Mp,ω , if p > 1, 2 −α/2 L f WMq,ω ≤ CIα f WMq,ω ≤ Cf M1,ω , if p 1, 2 8.4 where the constant C > is independent of f Property 8.1 is satisfied for large classes of differential operators We mention two of them − − a Consider a magnetic potential → a, that is, a real-valued vector potential → a a1 , a2 , , an , and an electric potential V We assume that for any k 1, 2, , n, ak ∈ Lloc and ≤ V ∈ Lloc The operator L, which is given by 2 − a V x, L − ∇ − i 8.5 is called the magnetic Schrodinger operator ă By the well-known diamagnetic inequality see 43, Theorem 2.3 we have the following pointwise estimate For any t > and f ∈ L2 , −tL e f ≤ e−tΔ f , 8.6 which implies that the semigroup e−tL has the kernel pt x, y which satisfies upper bound 8.1 b Let A aij x1≤i,j≤n be an n × n matrix with complex-valued entries aij ∈ L∞ satisfying Re n aij xζi ζj ≥ λ|ζ|2 i,j1 8.7 18 Journal of Inequalities and Applications for all x ∈ Rn , ζ ζ1 , ζ2 , , ζn ∈ Cn and some λ > Consider the divergence form operator Lf ≡ − div A∇f , 8.8 which is interpreted in the usual weak sense via the appropriate sesquilinear form It is known that the Gaussian bound 8.1 for the kernel of e−tL holds when A has realvalued entries see, e.g., 44, or when n 1, in the case of complex-valued entries see 45, Chapter 1 Finally we note that under the appropriate assumptions see 2, 46, Chapter 5; 45, pages 58-59 one can obtain results similar to Theorem 8.1 for a homogeneous elliptic operator L in L2 of order 2m in the divergence form Lf −1m Dα aαβ Dβ f |α||β|m 8.9 In this case estimate 8.1 should be replaced by pt x, y ≤ c3 t e−c4 |x−y|/t n/2m 1/2m 2m/2m−1 8.10 for all t > and all x, y ∈ Rn , where c3 , c4 > are independent of x, y, and t Acknowledgments The author thanks the referee for carefuly reading the paper and useful comments The author was partially supported by the Grant of the Azerbaijan-US Bilateral Grants Program II Project ANSF Award/AZM1-3110-BA-08 References 1 R R Coifman and Y Meyer, Au Delà des Opérateurs Pseudo-Différentiels, vol 57 of Astérisque, Société Mathématique de France, Paris, France, 1978 2 E M Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals, vol 43 of Princeton Mathematical Series, Princeton University Press, Princeton, NJ, USA, 1993 3 E M Stein and G Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Mathematical Series, no 3, Princeton University Press, Princeton, NJ, USA, 1971 4 V S Guliyev, Integral Operators on Function Spaces on the Homogeneous Groups and on Domains in Rn , Doctor’s dissertation, Steklov Mathematical Institute, Moscow, Russia, 1994 5 V S Guliyev, “Function spaces, integral operators and two weighted inequalities on homogeneous groups Some applications,” Baku, pp 1–332, 1999 Russian 6 V I Burenkov and H V Guliyev, “Necessary and sufficient conditions for boundedness of the maximal operator in local Morrey-type spaces,” Studia Mathematica, vol 163, no 2, pp 157–176, 2004 7 V I Burenkov and V S Guliyev, “Necessary and sufficient conditions for the boundedness of the Riesz potential in local morrey-type spaces,” Potential Analysis, vol 31, no 2, pp 1–39, 2009 8 V I Burenkov, V S Guliev, and G V Guliev, “Necessary and sufficient conditions for the boundedness of the Riesz potential in local Morrey-type spaces,” Doklady Mathematics, vol 75, no 1, pp 103–107, 2007, Translated from Doklady Akademii Nauk, vol 412, no 5, pp 585–589, 2007 9 V I Burenkov, H V Guliyev, and V S Guliyev, “Necessary and sufficient conditions for the boundedness of fractional maximal operators in local Morrey-type spaces,” Journal of Computational and Applied Mathematics, vol 208, no 1, pp 280–301, 2007 Journal of Inequalities and Applications 19 10 V I Burenkov, V S Guliev, T V Tararykova, and A Sherbetchi, “Necessary and sufficient conditions for the boundedness of genuine singular integral operators in Morrey-type local spaces,” Doklady Akademii Nauk, vol 422, no 1, pp 11–14, 2008 11 H G Eridani, “On the boundedness of a generalized fractional integral on generalized Morrey spaces,” Tamkang Journal of Mathematics, vol 33, no 4, pp 335–340, 2002 12 V S Guliev and R Ch Mustafaev, “Integral operators of potential type in spaces of homogeneous type,” Doklady Akademii Nauk, vol 354, no 6, pp 730–732, 1997 13 V S Guliev and R Ch Mustafaev, “Fractional integrals in spaces of functions defined on spaces of homogeneous type,” Analysis Mathematica, vol 24, no 3, pp 181–200, 1998 14 K Kurata, S Nishigaki, and S Sugano, “Boundedness of integral operators on generalized Morrey spaces and its application to Schrodinger operators,” Proceedings of the American Mathematical Society, ă vol 128, no 4, pp 11251134, 2000 15 L Liu, “Weighted inequalities in generalized Morrey spaces of maximal and singular integral operators on spaces of homogeneous type,” Kyungpook Mathematical Journal, vol 40, no 2, pp 339– 346, 2000 16 Sh Lu, D Yang, and Z Zhou, “Sublinear operators with rough kernel on generalized Morrey spaces,” Hokkaido Mathematical Journal, vol 27, no 1, pp 219–232, 1998 17 T Mizuhara, “Boundedness of some classical operators on generalized Morrey spaces,” in Harmonic Analysis (Sendai, 1990), S Igari, Ed., ICM-90 Satellite Conference Proceedings, pp 183–189, Springer, Tokyo, Japan, 1991 18 E Nakai, “Hardy-Littlewood maximal operator, singular integral operators and the Riesz potentials on generalized Morrey spaces,” Mathematische Nachrichten, vol 166, pp 95–103, 1994 19 E Nakai, “The Campanato, Morrey and Holder spaces on spaces of homogeneous type, Studia ă Mathematica, vol 176, no 1, pp 1–19, 2006 20 L Softova, “Singular integrals and commutators in generalized Morrey spaces,” Acta Mathematica Sinica, vol 22, no 3, pp 757–766, 2006 21 M Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, vol 105, Princeton University Press, Princeton, NJ, USA, 1983 22 A Kufner, O John, and S Fuˇc´ık, Function Spaces, Noordhoff International, Leyden, The Netherlands, 1977 23 C B Morrey Jr., “On the solutions of quasi-linear elliptic partial differential equations,” Transactions of the American Mathematical Society, vol 43, no 1, pp 126–166, 1938 24 F Chiarenza and M Frasca, “Morrey spaces and Hardy-Littlewood maximal function,” Rendiconti di Matematica e delle sue Applicazioni, vol 7, no 3-4, pp 273–279, 1987 25 J Peetre, “On convolution operators leaving Lp,λ spaces invariant,” Annali di Matematica Pura ed Applicata, vol 72, pp 295–304, 1966 26 D R Adams, “A note on Riesz potentials,” Duke Mathematical Journal, vol 42, no 4, pp 765–778, 1975 27 G Di Fazio and M A Ragusa, “Interior estimates in Morrey spaces for strong solutions to nondivergence form equations with discontinuous coefficients,” Journal of Functional Analysis, vol 112, no 2, pp 241–256, 1993 28 N K Bari and S B Steˇckin, “Best approximations and differential properties of two conjugate functions,” Trudy Moskovskogo Matematiˇceskogo Obˇscˇ estva, vol 5, pp 483–522, 1956 Russian ˇ Muhtarov, Introduction to the Theory of Nonlinear Singular Integral Equations, 29 A I Guse˘ınov and H S Nauka, Moscow, Russia, 1980 30 N K Karapetiants and N G Samko, “Weighted theorems on fractional integrals in the generalized Holder spaces H0ω ρ via the indices mω and Mω , Fractional Calculus & Applied Analisys, vol 7, no ă 4, pp 437–458, 2004 31 N Samko, “Singular integral operators in weighted spaces with generalized Holder condition, ă Proceedings of A Razmadze Mathematical Institute, vol 120, pp 107–134, 1999 32 N Samko, “On non-equilibrated almost monotonic functions of the Zygmund-Bary-Stechkin class,” Real Analysis Exchange, vol 30, no 2, pp 727–745, 2004-2005 33 V M Kokilashvili and S G Samko, “Operators of harmonic analysis in weighted spaces with nonstandard growth,” Journal of Mathematical Analysis and Applications, vol 352, no 1, pp 15–34, 2009 34 N G Samko, S G Samko, and B G Vakulov, “Weighted Sobolev theorem in Lebesgue spaces with variable exponent,” Journal of Mathematical Analysis and Applications, vol 335, no 1, pp 560–583, 2007 35 C L Fefferman, “The uncertainty principle,” Bulletin of the American Mathematical Society, vol 9, no 2, pp 129–206, 1983 20 Journal of Inequalities and Applications 36 Z W Shen, “Lp estimates for Schrodinger operators with certain potentials, Annales de lInstitut ă Fourier, vol 45, no 2, pp 513–546, 1995 37 J P Zhong, Harmonic Analysis for Some Schrăodinger type operators, Ph.D thesis, Princeton University, Princeton, NJ, USA, 1993 38 K Kurata and S Sugano, “A remark on estimates for uniformly elliptic operators on weighted Lp spaces and Morrey spaces,” Mathematische Nachrichten, vol 209, pp 137–150, 2000 39 S Sugano, “Estimates for the operators V α −Δ V −β and V α ∇−Δ V −β with certain nonnegative potentials V ,” Tokyo Journal of Mathematics, vol 21, no 2, pp 441–452, 1998 40 G Z Lu, “A Fefferman-Phong type inequality for degenerate vector fields and applications,” PanAmerican Mathematical Journal, vol 6, no 4, pp 37–57, 1996 sur les groupes nilpotents,” Journal of 41 H.-Q Li, Estimations Lp des operateurs de Schrodinger ă Functional Analysis, vol 161, no 1, pp 152–218, 1999 42 X T Duong and L X Yan, “On commutators of fractional integrals,” Proceedings of the American Mathematical Society, vol 132, no 12, pp 3549–3557, 2004 43 B Simon, “Maximal and minimal Schrodinger forms,” Journal of Operator Theory, vol 1, no 1, pp ă 3747, 1979 44 W Arendt and A F M ter Elst, “Gaussian estimates for second order elliptic operators with boundary conditions,” Journal of Operator Theory, vol 38, no 1, pp 87–130, 1997 45 P Auscher and P Tchamitchian, Square Root Problem for Divergence Operators and Related Topics, vol 249 of Astérisque, Société Mathématique de France, Paris, France, 1998 46 A McIntosh, “Operators which have an H1 -functional calculus,” in Miniconference on Operator Theory and Partial Differential Equations (North Ryde, 1986), vol 14 of Proceedings of the Centre for Mathematics, pp 210–231, The Australian National University, Canberra, Australia, 1986